On the radical of a group algebra over a commutative ring

Abstract

Several people, including Wallace [4] and Passman [3], have studied the Jacobson radical of the group algebra F[G] where F is a field and G is a multiplicative group. In [4], for instance, Wallace proves that if G is an abelian group with Sylow p-subgroup P and if F is a field of characteristic p, then the Jacobson radical of F[G] equals the right ideal generated by the radical of F[P]. In this paper we shall study group algebras over arbitrary commutative rings. By a reduction to the case of a semi-simple commutative ring, we obtain Theorem 1 whose corollary contains a generalization of Wallace's theorem. Theorem 2, on the other hand, uses the first theorem to obtain results related to the main theorem of [3].